We present a variationally consistent method for deriving residual-based closure models for incompressible Navier–Stokes equations. The method is based on the fine-scale variational structure facilitated by the variational multiscale framework where fine scales are driven by the residuals of the Euler–Lagrange equations of the resolved scales in the balance of momentum and conservation of mass equations. A bubble-functions based approach is applied directly to the fine-scale variational equation to derive analytical expressions for the closure model. Variational consistency of the model lends itself to rigorous linearization that results in quadratic rate of convergence of the method in the iterative solution strategy for the nonlinear equations. The method is shown to work for a family of linear and quadratic hexahedral and tetrahedral elements as well as composite discretizations that are comprised of hexahedral and tetrahedral elements in the same computational domain. Numerical tests with the proposed model are presented for various classes of turbulent flow problems to show its generality and range of applicability. The test cases investigated include Taylor–Green vortex stretching, statistically stationary wall-bounded channel flows, and modeling the effects of the geometry of the leading edge of the plate on the instability of the boundary layer that leads to flow separation and flow reversal over flat plates of finite thickness.

中文翻译：

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不可压缩湍流大涡模拟的变分导出闭合模型

我们提出了一种用于推导不可压缩 Navier-Stokes 方程的基于残差的闭合模型的变分一致方法。该方法基于由变分多尺度框架促进的精细尺度变分结构，其中精细尺度由动量平衡和质量方程守恒中解析尺度的欧拉-拉格朗日方程的残差驱动。基于气泡函数的方法直接应用于精细尺度变分方程，以推导出闭合模型的解析表达式。模型的变分一致性有助于严格的线性化，从而在非线性方程的迭代求解策略中导致该方法的二次收敛速度。该方法被证明适用于线性和二次六面体和四面体单元以及由同一计算域中的六面体和四面体单元组成的复合离散化。对各种类别的湍流问题提出了使用所提出模型的数值测试，以显示其普遍性和适用范围。研究的测试案例包括泰勒-格林涡旋拉伸、统计平稳的壁面边界通道流动，以及模拟板前缘几何形状对边界层不稳定性的影响，边界层的不稳定性导致流动分离和平面上的流动反转有限厚度的板。对各种类别的湍流问题提出了使用所提出模型的数值测试，以显示其普遍性和适用范围。研究的测试案例包括泰勒-格林涡旋拉伸、统计平稳的壁面边界通道流动，以及模拟板前缘几何形状对边界层不稳定性的影响，边界层的不稳定性导致流动分离和平面上的流动反转有限厚度的板。对各种类别的湍流问题提出了使用所提出模型的数值测试，以显示其普遍性和适用范围。研究的测试案例包括泰勒-格林涡旋拉伸、统计平稳的壁面边界通道流动，以及模拟板前缘几何形状对边界层不稳定性的影响，边界层的不稳定性导致流动分离和平面上的流动反转有限厚度的板。